Create CovarianceBinary.md

Author: maxbiostat

ProgrammingProjects/CovarianceBinary.md

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+## Efficiently computing covariance matrices for binary data  
+
+Let **X** be P-dimensional binary vector. Now suppose you have a sample **D** = (**X1**, **X2**, ..., **Xn**). The task is to compute the P x P covariance matrix of **X** from the sample. 
+
+In R you could presumably do
+```R
+cov(D) 
+```
+The problem is that in many cases this will give you a matrix that is not positive-definite. One way to fix the problem is to realise that for binary variable we can compute the covariance between Xi and Xj by computing
+```
+cov_ij = p_ij - p_i*p_j.
+```
+The problem then becomes doing this efficiently. The task is embarrassingly parallelisable, but coding in R is still slow.
+
+Your job is to take the implementation  of `binary_cov_matrix()` in [here](https://github.com/maxbiostat/BinaryMarkovChains/blob/main/R/binary_multiESS.R) and make it go vrum vrum. I reckon a simple re-coding in Rcpp should do the trick.
+
+**Applications**: this can be used in estimating the efficiency of Markov chain Monte Carlo algorithms in binary spaces.